A Production Inventory Model with Shortages, Fuzzy Preparation Time and Variable Production and Demand


A production inventory model is formulated for a single item. Here, demand varies with the on-hand inventory level and production price. Shortages are allowed and fully backlogged. The time gap between the decision and actual commencement of production is termed as “preparation time” and is assumed to be crisp/imprecise in nature. The set-up cost depends on preparation time. The fuzzy preparation time is reduced to a crisp interval preparation time using nearest interval approximation and following the interval arithmetic, the reduced problem is converted to a multi-objective optimization problem. Mathematical analysis has been made for single objective crisp model (Model-I). Numerical illustration have been made for both crisp (Model-I) and fuzzy (Model-II) models. Model-I is solved by generalized reduced gradient technique and multi-objective model (Model-II) by Global Criteria Method. Sensitivity analyses have been made for some parameters of Model-I.

Share and Cite:

N. Mahapatra, U. Bera and M. Maiti, "A Production Inventory Model with Shortages, Fuzzy Preparation Time and Variable Production and Demand," American Journal of Operations Research, Vol. 2 No. 2, 2012, pp. 183-192. doi: 10.4236/ajor.2012.22021.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] F. Harris, “Operations and Cost (Factory Management Series),” A.W. Shaw Co., Chicago, 1915, pp. 48-52.
[2] G. Hadley and T. M. Whitin, “Analysis of Inventory Systems,” Prentice Hall, Englewood Cliffs, 1963.
[3] R. J. Tersine, “Principles of Inventory and Materials Management,” Elsevier North Holland Inc., New York, 1982.
[4] E. A. Silver and R. Peterson, “Decision System for Inventory Management and Research,” John Wiley, New York, 1985.
[5] B. N. Mandal and S. Phaujdar, “A Note on Inventory Model with Stock-Dependent Consumption Rate,” OPSEARCH, Vol. 26, 1989, pp. 43-46.
[6] T. L. Urban, “Deterministic Inventory Models Incorporating Marketing Decisions,” Computer and Industrial Engineering, Vol. 22, No. 1, 1992, pp. 85-93. doi:10.1016/0360-8352(92)90035-I
[7] A. K. Bhunia and M. Maiti, “An Inventory Model for Decaying Items with Selling Price, Frequency of Advertisement and Linearly Time Dependent Demand with Shortages,” IAPQR Transactions, Vol. 22, 1997, pp. 41- 49.
[8] E. Naddor, “Inventory System,” John Wiley, New York, 1966.
[9] D. Magson, “Stock Control When Lead-Time Can Not Be Considered Constant,” Journal of the Operational Research Society, Vol. 30, 1979, pp. 317-322.
[10] B. Foote, N. Kebriaci and H. Kumin, “Heuristic Policies for Inventory Ordering Problems with Long and Random Varying Lead Times,” Journal of Operations Management, Vol. 7, No. 3-4, 1988, pp. 115-124. doi:10.1016/0272-6963(81)90008-5
[11] N. K. Mahapatra and M. Maiti, “Inventory Model for Breakable Item with Uncertain Preparation Time,” Tamsui Oxford Journal of Management Sciences, Vol. 20, No. 2, 2004, pp. 83-102.
[12] N. K. Mahapatra and M. Maiti, “Production-Inventory Model for a Deteriorating Item with Imprecise Preparation Time for Production in Finite Time Horizon,” Asia Pacific Journal of Operations Research, Vol. 23, No. 2, 2006, pp. 171-192.
[13] S. S. Rao, “Multi Objective Optimization in Structural Design with Uncertain Parameters and Stochastic Process,” AIAA Journal, Vol. 22, 1984.
[14] L. Li and K. K. Lai, “A Fuzzy Approach to the Multiobjective Transportation Problem,” Computers & Operations Research, Vol. 27, No. 1, 2000, pp. 43-57. doi:10.1016/S0305-0548(99)00007-6
[15] G. Padmanabhan and P. Vrat, “EOQ Models for Perishable Items under Stock-Dependent Selling Rate,” European Journal of Operations Research, Vol. 86, No. 2, 1995, pp. 281-292. doi:10.1016/0377-2217(94)00103-J
[16] T. K. Roy and M. Maiti, “Multi-Objective Inventory Models of Deteriorating Items With Some Constraints in a Fuzzy Environment,” Computers and Operations Research, Vol. 25, No. 12, 1998, pp. 1085-1095. doi:10.1016/S0305-0548(98)00029-X
[17] N. K. Mahapatra, T. K. Roy and M. Maiti, “Multi-Objective Multi-Item Inventory Problem,” Proceedings of the Seminar on Recent Trends and Developments in Applied Mathematics, Howrah, 3 March 2001, pp. 44-68.
[18] N. K. Mahapatra, K., Das, A. K. Bhunia and M. Maiti, “Multiobjective Inventory Model of Deteriorating Items With Ramp Type Demand Dependent Production, Setup and Unit Costs,” Proceedings of the National Symposium on Recent Advances of Mathematics and Its Applications in Science and Society, University of Kalyani, Kalyani, 21-22 November 2002, pp. 89-110.
[19] K. M. Miettinen, “Non-Linear Multi-Objective Optimization,” Kluwer’s International Series, Boston, 1999.

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.